The opposite number of a number $a$ is such a number $-a$ , that $$-a\!+\!a = 0.$$

Some properties:

• $-0 = 0$
• $-(-a) = a$
• $-(a\!+\!b) = (-a)\!+\!(-b)$
• $-(a\!\cdot\!b) = a\!\cdot\!(-b) = (-a)\!\cdot\!b$
• $-(a\!-\!b) = b\!-\!a$
• $-\sum_{j = 1}^n a_j = \sum_{j = 1}^n (-a_j)$
• $-\int_a^b f(x)\,dx = \int_b^a f(x)\,dx$

Corollary. If one changes the sign of one factor of a ring product, then the sign of the whole product changes.